BiBa — One Time Signature

You are at a fairground, and there’s a fun challenge. In front of you are five bins, and the challenge is for you to fire a ball at the bins, and where they will fall randomly into one of the bins. If you get two balls in the bin, you will win, but every attempt will cost you $1. The prize money is $4. Is it a good bet?

Okay. You have five bins, and you want to get to the point where one bin has two balls. For this you will have a 1-in-5 probability that any throw of the ball will go into a specific bin. What is the likely number of throws that it will take to win? Well, as a maximum will be six throws (you will lose $6, but gain $4, and will be thus $2 down). As a minimum it will take you two throws (for a $2 stake you will win $4). Will you win or will the fairground win, overall?

Of course, the probability on the first throw will be:

P_0 = 0

on the second we now have a one-in-5 chance of getting in the bin we hit the first time, so the chances of us not winning will be:

P_1 (Not win on second throw) = 4/5

Now, on the third throw, if we have not won, we will have two bins with balls out of 5 bins. So the bins could be:

1 1 0 0 0
1 0 1 0 0
1 0 0 1 0
1 0 0 0 1
0 1 1 0 0
0 1 0 1 0
0 1 0 0 1
0 1 0 1 0
0 1 0 0 1
You have three spaces that we could hit in each of the bins. You thus have two chances to hit a bin with a ball, and three chances to miss. The chance of you not winning is 3/5. The probability you will not…